The orientable exact Lagrangian fillability conjecture for twist-spun torus links

Let $k,n,\ell$ and $d$ be as in the construction of the twist-spun torus link, and let $\Sigma_{\psi^\ell}(\Lambda(k,n-k))$ denote that twist-spun link. An orientably exact Lagrangian filling is an orientable exact Lagrangian filling of this link. Fillability conjecture. The twist spun $\Sigma_{\psi^\ell}(\Lambda(k,n-k))$ is orientably exact Lagrangian fillable if and only if $k$ is congruent to $-1$, $0$, or $1$ modulo $d$.

This conjecture is proposed as a converse to the paper's filling theorem and as a generalization of a conjecture of Hughes. The preceding discussion indicates that cluster-theoretic obstructions may rule out fillings outside these congruence classes; the full equivalence remains open.